The formula a=23.1e^.0153t models the population of a us state, a, in millions, t years after 2000
a. what was the population of the state in 2000?
b. when will the population of the state reach 28.3 million?

We want to study the population formula of a given US state.

a) 23.1 millions.

b) in 2013 - 2014

The population state is:

A = 23.1*e^(0.0153*t)

In millions.

Where t represents the number of years after 2000.

a) For the population at the year 2000, we need to evaluate the function at t = 0 (2000 is 0 years after 2000).

A(0) = 23.1*e^(0.0153*0) = 23.1

The population at year 2000 was 23.1 millions.

b) Now we want to solve:

A(t) = 28.3 = 23.1*e^(0.0153*t)

         28.3/23.1 = e^(0.0153*t)

         1.23 = e^(0.0153*t)

Now we can apply the natural logarithm to both sides:

ln(1.23) = ln( e^(0.0153*t) )

ln(1.23) = 0.0153*t

ln(1.23)/0.0153 = t = 13.5

So 13.5 years after 2000 the population will be 28.3 million.

If you want to learn more, you can read:

link

Part A

Plug t = 0 into the function to find the population in the year 2000.

This works because t represents the number of years since 2000.

So t = 0 represents 2000, t = 1 is 2001, t = 2 is 2002, and so on.

A = 23.1*e^(0.0153*t)

A = 23.1*e^(0.0153*0)

A = 23.1*e^(0)

A = 23.1*1

A = 23.1

Recall that the variable A is the population in millions.

Saying A = 23.1 is saying there are 23.1 million people (ie 23 million, 100 thousand)

23.1 million

Part B

Replace A with 28.3 and use natural logarithms to isolate t.

We'll use the log rules Ln(e) = 1 and Ln(x^y) = y*Ln(x).

A = 23.1*e^(0.0153*t)

28.3 = 23.1*e^(0.0153*t)

23.1*e^(0.0153*t) = 28.3

e^(0.0153*t) = 28.3/23.1

e^(0.0153*t) = 1.22510822510822

Ln[ e^(0.0153*t) ] = Ln[ 1.22510822510822 ]

0.0153*t*Ln[ e ] = Ln[ 1.22510822510822 ]

0.0153*t*1 = Ln[ 1.22510822510822 ]

0.0153*t = Ln[ 1.22510822510822 ]

t = Ln[ 1.22510822510822 ]/0.0153

t = 13.2698815112052

It will take approximately 13.2699 years for the population to reach the desired target.

This result is between 13 and 14.

Between 2013 and 2014

its a straight line parallel to the time axis